OFFSET
1,1
COMMENTS
In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribing circle, and the vertices are said to be concyclic.
The area A of any cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula: A = sqrt((s - a)(s - b)(s - c)(s - d)) where s, the semiperimeter, is s= (a+b+c+d)/2.
The circumradius R (the radius of the circumcircle) of any cyclic quadrilateral is given by
R = sqrt((ab+cd)(ac+bd)(ad+bc))/(4A).
Many cyclic quadrilaterals [a, b, c, d] with integer sidelengths, integer area, and integer circumradius have the property that a = b and c = d, thus forming a kite with two right angles, with the long diagonal of the kite being a diameter of the circle; thus the circumradius is R = sqrt(a^2 + c^2)/2. Since Brahmagupta's formula is invariant upon permutation of the sides, the area of such a kite is the same as that of the rectangle with sides [a, c, b, d]. So in this case s = a+c, and A = a*c. In particular, the double of any Pythagorean triple will satisfy our requirements.
Nevertheless, there also exist cyclic quadrilaterals with integer sidelengths, integer area, and integer circumradius, whose four sides are distinct; for example, [a, b, c, d] = [ 14, 30, 40, 48] => A = 936 and R = 25.
LINKS
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
EXAMPLE
(a(1),a(2)) = (6,8) because, for (a,b,c,d) = (6,6,8,8) we obtain:
s = a + c = 6+8 = 14;
A = a*c = 6*8 = 48;
R = sqrt(a^2 + c^2)/2 = sqrt(6^2 + 8^2)/2 = 5.
MATHEMATICA
nn=1500; lst={}; Do[s=(2*a+2*c)/2; If[IntegerQ[s], area2=(s-a)^2*(s-c)^2; If[0<area2&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[(a^2+c^2)*4*a^2*c^2/((s-a)^2*(s-c)^2)]/4], AppendTo[lst, Union [{a}, {c}]]]], {a, nn}, {c, a}]; Union[lst]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, May 22 2014
EXTENSIONS
Definition and comments extended and/or corrected by Gregory Gerard Wojnar, Nov 10 2018
STATUS
approved